[seqfan] Re: formula for Pi
Robert Israel
israel at math.ubc.ca
Sun Dec 7 20:27:57 CET 2008
This series has no simple relation to the digits of Pi (whether base
10 or any other base). Moreover, it converges quite slowly, since the
terms are of order 1/n^2. By the way, it follows quite easily from the
identity
sum_{n=0}^infty (1/(n+b) - 1/(n+a)) = Psi(a) - Psi(b)
together with
Psi(5) = 25/12-gamma,
Psi(22/3) = 699171/138320-gamma-1/6*Pi*3^(1/2)-3/2*ln(3),
Psi(7/3) = 15/4-gamma-1/6*Pi*3^(1/2)-3/2*ln(3)
noting that the summand is 22/(n+7/3) - 7/(n+5) - 15/(n+22/3).
A nicer version, in my opinion, is
sum_{n=0}^infty 1/((3*n+1)*(3*n+2)) = Pi*sqrt(3)/9
Cheers,
Robert Israel
On Sun, 7 Dec 2008, Hector Zenil wrote:
> Is it possible to get subsequences of the digits of Pi from this
> formula? Of course without doing N[formula,digits] or any other
> similar trick...
>
>
> On Sun, Dec 7, 2008 at 3:28 PM, Vladimir Bondarenko <vb at cybertester.com> wrote:
>> Hello,
>>
>> This is an exact formula for Pi.
>>
>> FullSimplify[6/7*(1/3*Sum[(843*n + 4607)/((n + 5)*(3*n + 7)*(3*n + 22)),
>> {n, 0, Infinity}] - 655999/248976 - 7/2*Log[3])*Sqrt[3]]
>>
>> Pi
>>
>> Cheers,
>>
>> Vladimir Bondarenko
>>
>> Quoting Alexander Povolotsky <apovolot at gmail.com>:
>>
>>> I've got this very ugly formula by playing Maple syntax via old and
>>> new inverse symbolic calculators :
>>>
>>> Pi = 6/7*(1/3*sum((843*n + 4607)/((n+5)*(3*n+7)*(3*n+22)),n=0...infinity)
>>> - 655999/248976 - 7/2*ln(3))*sqrt(3)
>>>
>>> What I've got for Pi above - is it just a good approximation or exact ?
>>> (My old PC with PARI/GP can not get over the summing )
>>>
>>> It looks that this 655999/248976 fraction quickly becomes periodic around
>>> 920249341301972880
>>>
>>> gp > \p 1000
>>> realprecision = 1001 significant digits (1000 digits displayed)
>>> gp > 1.0*655999/248976
>>> %4 = 2.6347880
>>> 920249341301972880
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>>> 92024934130197......
>>> 288092....
>>>
>>> Cheers Alexander R. Povolotsky
>>>
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>
>>
>>
>>
>>
>> _______________________________________________
>>
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>>
>
>
>
> --
> Hector Zenil http://www.mathrix.org
>
>
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